59.1.700 problem 717

Internal problem ID [9872]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 717
Date solved : Sunday, March 30, 2025 at 02:48:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

ode:=diff(diff(u(x),x),x)+4/x*diff(u(x),x)-a^2*u(x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = \frac {c_1 \,{\mathrm e}^{a x} \left (a x -1\right )+c_2 \,{\mathrm e}^{-a x} \left (a x +1\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 68

ode=D[u[x],{x,2}]+4/x*D[u[x],x]-a^2*u[x]==0; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\[ u(x)\to \frac {\sqrt {\frac {2}{\pi }} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{a x^{5/2} \sqrt {-i a x}} \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
a = symbols("a") 
u = Function("u") 
ode = Eq(-a**2*u(x) + Derivative(u(x), (x, 2)) + 4*Derivative(u(x), x)/x,0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = \frac {C_{1} J_{\frac {3}{2}}\left (x \sqrt {- a^{2}}\right ) + C_{2} Y_{\frac {3}{2}}\left (x \sqrt {- a^{2}}\right )}{x^{\frac {3}{2}}} \]